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Representing smooth 4-manifolds as loops in the pants complex

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Klug,  Michael
Max Planck Institute for Mathematics, Max Planck Society;

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1912.02325.pdf
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Citation

Islambouli, G., & Klug, M. (2021). Representing smooth 4-manifolds as loops in the pants complex. Mathematical Research Letters, 28(6), 1703-1738. doi:10.4310/MRL.2021.v28.n6.a4.


Cite as: https://hdl.handle.net/21.11116/0000-000B-6725-8
Abstract
We show that every smooth, orientable, closed, connected 4-manifold can be
represented by a loop in the pants complex. We use this representation,
together with the fact that the pants complex is simply connected, to provide
an elementary proof that such 4-manifolds are smoothly cobordant to $\coprod_m
\mathbb{C}P^2 \coprod_n \bar{\mathbb{C}P}^2$. We also use this association to
give information about the structure of the pants complex. Namely, given a loop
in the pants complex, $L$, which bounds a disk, $D$, we show that the signature
of the 4-manifold associated to $L$ gives a lower bound on the number of
triangles in $D$.